BQ# 7: Unit V

 The origins of the the difference quotient comes from a graph and using an old equation from early this year (Unit A).

Here is the graph f(x) and the line that's barely touching it at x is the tangent line. We can write the coordinates for the graph as ( x, f(x) ). 

This is another graph that has a secant line going through it. It still has the original point as the first one but it has another point in it. Since we deviated from the original to the new point, then it's a change in the graph. That change can be written as delta x or as i put it h. So the new coordinates for this graph would be ( x, f(x) ), ( x+h, f(x+h) ).

Here is an image with the one of the equations, the difference quotient, we would use to find the equation. First we take the coordinates from the graph, then we use the slope formula to ind the slope,Then we plug in the denominator we see that the xs' cancel so there's only an  h left in the denominator. The last one in purple is the difference quotient and that is how we get the equation.